Available online at J. Math. Comput. Sci. 4 (2014), No. 4, ISSN:

Transcription

1 Available olie a hp://sci.org J. Mah. Compu. Sci. 4 (2014), No. 4, ISSN: ON ITERATIVE TECHNIQUES FOR NUMERICAL SOLUTIONS OF LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS S.O. EDEKI *, A.A. OPANUGA, H.I. OKAGBUE Deparme of Mahemaics, College of Sciece & Techology, Covea Uiversiy, Nigeria Copyrigh 2014 Edei, Opauga ad Oagbue. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial wor is properly cied. Absrac: This paper preses Differeial Trasformaio Mehod (DTM) ad Picard s Ieraive Mehod () as compuaioal echiques i solvig liear ad oliear differeial equaios. For umerical aalysis of he mehods, hree examples are cosidered. The resuls obaied are compared wih heir correspodig exac soluios. A li bewee successive erms of he soluios usig he wo mehods is oed. The DTM is very effecive ad reliable i obaiig approximae soluios. The requires he saisfacio of Lipschiz coiuiy codiio; hough, is resuls also coverge rapidly o he exac soluios. Keywords: differeial rasform; Picard s ieraio; differeial equaio; Lipschiz cosa AMS Subjec Classificaio: 35F25, 34K28, 35C Iroducio May aalyical, semi-aalyical or purely umerical mehods are available for he soluio of differeial equaios ecouered i maageme scieces, pure ad applied scieces. Mos of hese mehods are compuaioally iesive because hey are rial-error i aure, or eed complicaed symbolic compuaios [1]. Youssef used Picard ieraio echique wih Gauss-seidel echique for iiial value problem [2], Rach used Adomia Decomposiio mehod ad Picard s mehod [3]. Bellomo * Correspodig auhor Received Jue 10,

2 ON ITERATIVE TECHNIQUES 717 ad Sarafya also compared Adomia Decomposiio mehod ad Picard ieraive scheme [4]. The differeial rasformaio is a umerical mehod for solvig differeial equaios. The cocep of differeial rasform was firs iroduced by Zhou (1986) while solvig liear ad o-liear iiial value problems i elecric circui aalysis [5]. Che ad Liu applied his mehod o solve wo-boudary problems [6]. Jag e al, apply he wo-dimesioal differeial mehod o solve parial differeial equaios [7]. Edei e al [8] applied he differeial rasform mehod (DTM) as a semi-aalyical mehod o a cerai class of ODEs. The DTM has bee applied o oher areas amog- differece equaios [9], differeial-differece equaios [10], wo-dimesioal iegral equaios [11], opimizaio of he recagular fis wih variable hermal parameers [12] ad iegro-differeial equaios [13]. I his paper, liear ad oliear ordiary differeial equaios are cosidered usig he DTM ad he. The umerical resuls from he wo mehods are compared wih heir exac soluios. The mai advaage of he DTM is ha, i ca be applied direcly o liear ad oliear ordiary differeial equaios wihou liearizaio, discreizaio or perurbaio. Also, i is capable of grealy reducig he size of compuaioal wor while sill maiaiig accuracy, ad providig he series soluio wih fas covergece rae. The is also effecive bu requires he saisfacio of he Lipschiz coiuiy codiio. 2. Aalysis of he Basic Mehods I his secio, he basic coceps ad heorems for he Differeial Trasform Mehod (DTM) ad he Picard s Ieraive Mehod () are sysemaically iroduced. 2.1 The Fudameal of he Differeial Trasform Mehod Le y f ( x) be a arbirary fucio expressed i Taylor series abou a poi x 0 as x d f f( x)! dx (1) 0 x0 The, he differeial rasformaio of f( x ) is defied as

3 718 S.O. EDEKI, A.A. OPANUGA, AND H.I. OKAGBUE 1 d y F ( )! dx x 0 (2) As a resul, he iverse differeial rasform of F ( ) is: f ( x) x Y( ) (3) Special heorems of he DTM The followig heorems ca be deduced from equaios (1), (2) ad (3): Theorem 1: If y( x) y1( x) y2( x), he Y ( ) Y1( ) Y2( ) Theorem 2: If y( x) cy1( x), hey ( ) cy1 ( ), where c is a cosa. d y1 Theorem 3: If ( ) ( x ) ( )! yx, he Y ( ) Y 1( ) dx! Theorem 4: If y( x) y1( x) y2( x), he Y ( ) Y ( ) Y ( ) Theorem 5: If y( x) x, he Y( ) ( ) where ( ). 1, 2.3 Aalysis of he Picard Ieraio Mehod Cosider he firs order ordiary differeial equaio (IVP) 0, y g(, y), y( 0) y0 (4) To guaraee he exisece ad uiqueess of he soluio of (4), we assume ha g(, y ) is * Lipschiz coiuous i a ball, B ( ) b y 0 ; cere y 0 ad radius b. We defie a complee ormed space,,, g for he fucio g(, y ) equipped wih he sup-orm: g sup 0, T g, y( ) (5) where is a Hilber space,, a ier produc, ad a orm operaor w.r. v, v such ha: * g C a, b B ( y ) (6) a 0 b 0

4 ON ITERATIVE TECHNIQUES 719 where a * a, a ad B ( y ) y b, y b (7) b Thus, for every pair of pois y, y i Gr g ; he graph of g, here exiss a cosa M 0, such ha : g(, y ) g(, y ) M y y (8) where M is a Lipschiz cosa. Now by iegraig boh sides of (4) we ge: y d g(, y( )) d (9) 0 0 Thus, by fudameal heorem of calculus, (5) becomes: y y 0 g(, y( )) d 0 0 y y 0 g(, y( )) d (10) For a arbirary, i is obvious ha y appears boh o he LHS ad i he iegrad of (10). Therefore, we resor o ieraive approach (Picard) by choosig a iiial guess y y ad seig for 1, : 0 0 y 1 y0 g(, y ( )) d (11) 0 Thus, he approximae soluio o (4) is () 1 y, provided he limis i (11) exis such ha: 1 () lim 1( ) lim ( ) y y y (12) 3. Applicaios ad Numerical Resuls I his subsecio, we will cosider some differeial equaios (IVP) ad solve hem usig boh mehods- he differeial rasform mehod DTM ad he as discussed above.

5 720 S.O. EDEKI, A.A. OPANUGA, AND H.I. OKAGBUE Example1 Cosider he IVP: wih a exac soluio: Soluio (DTM): y y 1 0, y(0) 2 (13) y ( ) 1 e (14) ex We rewrie (13) i a sadard form ad ae he differeial rasform (DT) as follows; DT y( ) y( ) 1 By usig he basic ideas ad heorems of he DTM as saed above, we obai he followig recurrece relaio as follows: ( 1) Y( 1) Y( ) ( ), So, wih he iiial codiios Y(0) 2, 1 Y( 1) [ Y( ) ( )] 1 (15) Hece, for 0, we obai values for Y(1), Y(2), Y (3), as showed below: for 0, Y(1) 1; for 1, Y(2) ; for 2, Y(3) ; for 3, Y(4), 2! 4! 5! y( ) 2... (16) 2! 3! 4! 5!! DTM 4 ( ) 2. (17) 2! 3! 4! 5! Soluio (): We re-express (13) i a iegral form of (11) : y 1 y0 g(, y ( )) d 0 0 y 1 2 ( 1 y( )) d, 0 0, y0 2 (18) Hece, he followig successive approximaios are obaied:

6 ON ITERATIVE TECHNIQUES y0 2, y1 2, y2 2, y3 2, y4 2, 2! 2! 3! 2! 3! 4! y( ) 2 2! 3! 4!! S y( ) 2... (19) 2! 3! 4!! DTM ( ) 2 y ( ) (20) 2! 3! 0 Example 2 Cosider he IVP: wih a exac soluio: Soluio (DTM): y 2y 2, y(0) 0 (21) y ( ) 1 e ex 2 (22) We rewrie (21) i a sadard form ad ae he differeial rasform (DT) as follows; DT y( ) 2 y( ) 2, Y 0 0 ( 1) Y ( 1) 2 Y ( ) ( ), Y( 1) Y ( ) ( ) wih he iiial codiios Y(0) 2, 2 1 (23) Thus, for 0., we obai values for Y(1), Y(2), Y (3), as showed below: for 0, Y(1) 2; for 1, Y(2) ; for 2, Y(3) ; for 3, Y(4), 2! 3! 4! Hece, y( ) Y( ) 2 (24) 2! 3! 4! DTM 4 ( ) 2 (25) 2! 3! 4! 5! Soluio (): We re-express (21) i a iegral form of (11):

7 722 S.O. EDEKI, A.A. OPANUGA, AND H.I. OKAGBUE y 1 y0 g(, y ( )) d Hece, he followig successive approximaios are obaied: Thus, 0 0 y 1 2 (1 y( )) d, 0 0, y0 0 (26) (2 ) (2 ) (2 ) y0 0, y1 2, y2 2, y3 2, 2! 2! 3! (2 ) (2 ) (2 ) (2 ) 6 ( ) 2 (27) 2! 3! 4! 5! Firs order o-liear differeial equaios Example 3 Cosider he IVP: wih a exac soluio: Soluio (DTM): y y y (28) 2 ( ) ( ) 1, (0) 0 y ( ) a( ) ex (29) We rewrie (23) i a sadard form ad ae he differeial rasform (DT) as follows; DT y 2 1 y, ( 1) Y( 1) ( ) Y( r) Y( r), r0 1 Y( 1) ( ) Y( r) Y( r) 1 (30) r0 wih he iiial codiios Y(0) 0, Therefore, for 0, we obai values for Y(1), Y(2), Y (3), as showed below: 1 2 for 0, Y(1) 1; for 2, Y(3) ; for 4, Y(5) ; for 6, 3 15 where Y(0) Y(2) Y(4) Y(2 2) 0, for 1 Hece,

8 ON ITERATIVE TECHNIQUES 723 Soluio (): y( ) Y( ) (31) DTM 6 ( ). (32) 3 15 We re-express (28) i a iegral form of (11): y 1 y0 g(, y( )) d y y (1 y ( )) d, 0, y 0 (33) 0 Hece, he followig successive approximaios are obaied: y0 0, y1, y2, y3, As such, () (34) Remar 3.1: We observe a li bewee he soluios obaied usig he DTM ad he. This is expressed as: y DTM Y (35) Numerical Compariso of he exac soluio, he DTM soluio, ad he soluio I he subsecio, comparisos bewee he soluios for each example are displayed i he followig ables wih heir graphs i figures 1-3 respecively.

9 724 S.O. EDEKI, A.A. OPANUGA, AND H.I. OKAGBUE Table 1: Numerical compariso for Example 1 Exac soluio DTM 3 4 Absolue Error Absolue Error ( DTM) () E E Table 2: Numerical compariso for Example 2 Exac soluio DTM 5 6 Absolue Error Absolue Error ( DTM) () E E E E E

10 ON ITERATIVE TECHNIQUES 725 Table 3: Numerical compariso for Example 3 Exac soluio DTM 6 8 Absolue Error Absolue Error ( DTM) () E-05 2E Remar 3.1: We show i Figure [1-3], he graphs represeig he soluios of he solved examples. Series [1-3] idicae soluios for exac, DTM ad respecively. Fig 1: Graph of example 1 Soluios Fig 2: Graph of example 2 Soluios

11 726 S.O. EDEKI, A.A. OPANUGA, AND H.I. OKAGBUE Fig. 3: Graph of example 3 soluio 4.0 Discussio of Resuls ad Cocludig Remars I his paper, we have used he DTM ad he successfully i solvig boh liear ad oliear differeial equaios (IVP), ad he resuls obaied are compared wih heir correspodig exac soluios. I is observed ad oed ha all previous erms of he DTM are embedded i he correspodig sage of he. More accuracy is recorded as he umber of erms i he ieraios is icreased. Resuls from boh mehods coverge faser o heir exac soluios. The DTM rasforms he differeial equaios o algebraic-recursive equaios; hece, i is very effecive ad reduces he size of compuaioal wor wihou liearizaio, perurbaio or discreizaio of he give problem while he rasforms a differeial equaio o is equivale i iegral form provided he Lipschiz coiuiy codiio is saisfied. Coflic of Ieress The auhors declare ha here is o coflic of ieress.

12 ON ITERATIVE TECHNIQUES 727 REFERENCES [1] M.A. Mohamed, Compariso Differeial Trasformaio Techique wih Adomia Decomposiio mehod for Disperse Log-wave Equaios i (2+1) Dimesios, Applicaios ad Applied Mahemaics (AAM) 5 (2006), [2] I.K. Youssef, Picard ieraio algorihm combied wih Gauss-Seidel echique for iiial value problem, Applied Mahemaics ad compuaio, 190 (2007), [3] R. Rach, O he Adomia Decomposiio mehod ad comparisos wih Picard s mehod, J. Mah. Aal. Appl., 128 (1987), [4] N. Bellomo, ad D.Sarafya, O Adomia Decomposiio mehod ad some comparisos wih Picard s mehod. J. Mah. Aal. Appl., 123 (1987), [5] J.K. Zhou, Differeial Trasformaio ad is Applicaios for Elecrical circuis, Huazhog Uiversiy press, Wuha, Chia, 1986 (i Chiese). [6] C.L. Che, Y.C. Liu, Differeial Trasformaio Techique for seady oliear hea coducio problems. Appl. Mah. Comp. 95 (1998) [7] M.J. Jag, C.L. Chem, Aalysis of he respose of a srogly oliear damped sysem usig a Differeial Trasformaio Techique, Appl. Mah. Comp. 88 (1997) [8] S.O. Edei, H.I. Oagbue, A.A. Opauga, ad S.A. Adeosu (2014), A Semi-aalyical Mehod for Soluios of a Cerai Class of Ordiary Differeial Equaios. Applied Mahemaics, 5 (2014), hp://dx.doi.org/ /am [9] A. Arioglu ad I. Ozol, Soluio of differeial differece equaio by usig Differeial Trasformaio mehod, Appl. Mah. Compu, 173 (1) (2006), [10] A. Arioglu ad I. Ozol, Soluio of differeial differece equaio by usig Differeial Trasformaio mehod, Appl. Mah. Compu, 181 (1) (2006), [11] A. Tari,M. Rahimi, ad F. Talai, Solvig a class of 2-dimesioal liear ad oliear Volerra iegral equaios by Differeial Trasformaio mehod. J. Compu. Appl. Mah., 228 (2008), doi: /j. cam [12] L.T. Yu, C. K. Che, Applicio of Taylor rasformaio o opimize recagular fis wih variable hermal parameers, Appl. Mah. Model. 22 (1998), [13] A. Arioglu, ad I. Ozol, Soluio of boudary value problems for iegro-differeial equaios by usig Differeial Trasformaio mehod. Appl. Mah. Compu. 168 (2005),